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October, 1974 The Optimal Reward Operator in Dynamic Programming
D. Blackwell, D. Freedman, M. Orkin
Ann. Probab. 2(5): 926-941 (October, 1974). DOI: 10.1214/aop/1176996558

Abstract

Consider a dynamic programming problem with analytic state space $S$, analytic constraint set $A$, and semi-analytic reward function $r(x, P, y)$ for $(x, P)\in A$ and $y\in S$: namely, $\{r > a\}$ is an analytic set for all $a$. Let $Tf$ be the optimal reward in one move, with the modified reward function $r(x, P, y) + f(y)$. The optimal reward in $n$ moves is shown to be $T^n0$, a semi-analytic function on $S$. It is also shown that for any $n$ and positive $\varepsilon$, there is an $\varepsilon$-optimal strategy for the $n$-move game, measurable on the $\sigma$-field generated by the analytic sets.

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D. Blackwell. D. Freedman. M. Orkin. "The Optimal Reward Operator in Dynamic Programming." Ann. Probab. 2 (5) 926 - 941, October, 1974. https://doi.org/10.1214/aop/1176996558

Information

Published: October, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0318.49021
MathSciNet: MR359818
Digital Object Identifier: 10.1214/aop/1176996558

Subjects:
Primary: 49C99
Secondary: 28A05 , 60K99 , 90C99

Keywords: analytic sets , dynamic programming , gambling , optimal reward , optimal strategy

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 5 • October, 1974
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